Specializing cubulated relatively hyperbolic groups (with J.F. Manning). Preprint (2020). .pdf
In , Agol proved the Virtual Haken and Virtual Fibering Conjectures by confirming a conjecture of Wise: Every cubulated hyperbolic group is virtually special. We extend this result to cocompactly cubulated relatively hyperbolic groups with minimal assumptions on the parabolic subgroups. Our proof pro- ceeds by first recubulating to obtain an improper action with con- trolled stabilizers (a weakly relatively geometric action), and then Dehn filling to obtain many cubulated hyperbolic quotients. We apply our results to prove the Relative Cannon Conjecture for cer- tain cubulated or partially cubulated relatively hyperbolic groups.
One of our main results (Theorem A) recovers via different methods a theorem of Oregón-Reyes .